© The Authors 2025

1. Introduction

Strong gravitational lensing serves as a powerful tool in astrophysics, enabling the estimation of mass distribution without assuming the nature or status of the matter. This allows us to constrain the model of formation and evolution of the galaxy and galaxy cluster (Narayan & Bartelmann 1996; Schneider 2006). In lensing, multiple images can form when a background source is well aligned with the lens. The propagation of light along different trajectories introduces a relative time delay between the images, resulting from both the Shapiro effect and the difference in path length. It has been found that the lensing time delay can be used to estimate the Hubble constant (Refsdal 1964), providing a measurement independent of the cosmic-distance ladder or the cosmic microwave background. Constraints have been carried out from observations (e.g. Tewes et al. 2013; Suyu et al. 2014). In order to measure the time delay between multiple images, the background source must exhibit variability. Quasi-stellar objects (QSOs) are commonly used for this purpose due to their high number density in the Universe. However, because of their stochastic variability and long timescale, extensive monitoring is required to achieve an accurate measurement of the time delay. Fast radio bursts (FRBs) are a class of extragalactic radio transient with millisecond durations (e.g. Lorimer et al. 2007; Cordes & Chatterjee 2019). Thanks to their short duration and high brightness, FRBs offer a promising alternative for the background source in time-delay measurement, and to study cosmology and fundamental physics (e.g. Wucknitz et al. 2021).

In addition to gravity, plasma can deflect light (e.g. Perlick et al. 2015). The influence of plasma lensing depends on the frequency of light. Given the typical density of plasma in the interstellar or intergalactic medium (∼10⁻⁵−10⁻¹ electron cm⁻³), the effects of plasma lensing are minimal and detectable primarily at low frequencies (e.g. Bisnovatyi-Kogan & Tsupko 2010; Tuntsov et al. 2016). The deflection angle caused by plasma is of the order of 1 milliarcsec and generally negligible with regard to the image positions. However, variations in magnification or brightness and time delay can be significant and detectable, especially for the short-duration signal. For example, strong brightness variations observed in one repeating FRB have been ascribed to a plasma lensing event (Chen et al. 2024). Moreover, polarisation measurements provide critical insights into the emission mechanism of the sources and the properties of magnetic field along the line of sight. When light propagates through a magnetised plasma, the two circular polarisation modes experience different refractive indices due to their distinct propagation speeds. This phenomenon leads to rotation of the linear polarisation of the signal, known as the Faraday rotation. Plasma lensing in the presence of strong variations in the magnetic field can exhibit birefringence effects, which may produce intriguing observational signatures. For example, when the lensed image lies close to the critical curves, the polarisation state of the background source can even undergo a flip (Suresh & Cordes 2019; Gwinn 2019; Er et al. 2023). In extreme cases, the separation of the two polarisation modes can result in the splitting of the lensed images.

Birefringence can also occur in strongly lensed radio sources, with even cross-talk between gravitational lensing and plasma lensing. The propagation of light through magnetic plasma in a gravitational field has been explored in several studies (e.g. Breuer & Ehlers 1980, 1981a,b; Broderick & Blandford 2003). The dispersion relation exhibits complex behaviour in the presence of a gravitational field. In the case of a strong magnetic field, the lensed images of the two modes can even split as well (Turimov et al. 2019). In this work, we investigated the birefringence in strongly lensed radio sources, focusing on magnification and time delays. Additional lensed images can form, and, importantly, the distinct propagation paths of the two polarisation modes lead to different arrival times. As a result, the Faraday rotation relation cannot be used as an accurate estimate of the magnetic field. We demonstrate these effects using two toy examples: a galaxy scale lens with a weak magnetic field and a point lens with a strong magnetic field.

2. The deflection with magnetic field

We studied the influence of the magnetic field on the propagation of light in curved space time. We also explored several astrophysical scenarios, such as an FRB lensed by a galaxy or a galaxy cluster, where the magnetic field is relatively weak. Additionally, we consider cases where light propagates near compact objects, such as neutron stars or black holes, which are associated with strong magnetic field. In both scenarios, we restricted our analysis to static fields, i.e. where there is no temporal variation in the gravitational field, plasma density, or magnetic field.

In general, solving the Maxwell equations precisely in a relativistic medium is not feasible. Therefore, several approximations were employed in this work. First, we adopted the framework of Hamilton’s geometric theory of rays to study the optics in a curved space time with a magnetised medium. Second, we restricted our study to the weak gravitational field and small deflection angles, i.e. focusing exclusively on the far-field regime. For scenarios involving light deflection near a black hole or neutron star, for example to study the shadow of the black hole, one can find more details in Perlick & Tsupko (2022), among others. We applied the thin-lens approximation, as the lens size is small compared to the distance between the lens and the observer. Moreover, we modeled the electromagnetic waves as as plane, as well as monochromatic ones, with wavelengths significantly smaller than the characteristic scale of plasma variation (Breuer & Ehlers 1980; Broderick & Blandford 2003). Thus, the wave optical effects were not considered in this analysis (e.g. Jow et al. 2020; Feldbrugge 2023; Shi 2024).

We denote plasma frequency by ω_e¹ and the Larmor frequency by ω_B,

$${\omega }_{\mathrm{e}}^2(x)=\frac{4\pi n_{\mathrm{e}}(x)e^2}{m_{\mathrm{e}}}\equiv K_{\mathrm{e}}{n}_{\mathrm{e}}(x),{\omega }_B(x)=\frac{e}{m_{\mathrm{e}}}B(x)·\hat{k},$$(1)

where n_e(x) represents the number density of electrons in three dimensions, B(x) is the magnetic field, and $\hat{k}$ is the unit wave vector. Typically, both the plasma frequency and the Larmor frequency are low. For example, a plasma with electron density n_e = 1000 cm⁻³ has a frequency of only ∼0.3 MHz, and a magnetic field of B = 1 Gauss yields a Larmor frequency of about 2.8 MHz. These values are significantly lower than the typical observational frequencies (∼GHz), ensuring that ω≫ω_e,ω_B for most observations.

2.1. Nearly flat space time

The dispersion relation is critical for this study, particularly since the effect of the magnetic field in curved space time is complex. We adopted the results from Broderick & Blandford (2003, 2004), which use the locally flat centre-of-mass rest frame. The function D(k_μ,x^μ) is analogous to the Hamiltonian of the wave in the magnetic plasma (Breuer & Ehlers 1980, 1981b). We considered the far field, where the space can be approximated as nearly flat. We also applied the quasi-longitudinal approximation for the cold plasma, resulting in transverse wave modes. As a consequence, certain modes are absent when the light is deflected. However, this inaccuracy is acceptable for cases only involving small deflection angles (discussed further later). The dispersion relation is similar to that in the flat space time:

$$D=\left[g^{\mu \nu }{k}_{\mu }{k}_{\nu }+{\omega }_{\mathrm{e}}^2\pm {\omega }_{\mathrm{e}}^2\frac{{\omega }_B}{\omega (x)}\right].$$(2)

We emphasise that the observational frequency ω(x) is not a constant, it depends on the gravitational field: i.e. g₀₀². We neglect such variations since we focus on the weak field cases. Then we can solve for the ray equations by

$$\frac{{\mathit{dx}}^{\nu }}{d\tau }=\frac{\partial D}{\partial k_{\nu }},\frac{d{k}_{\nu }}{d\tau }=-\frac{\partial D}{\partial x^{\nu }},$$(3)

where τ is an affine parameter. The refractive indices can be written as

$$n_{L,R}=1-\frac{\omega {\omega }_{\mathrm{e}}^2}{2{\omega }^2(\omega \pm {\omega }_B)}\approx 1-\frac{{\omega }_{\mathrm{e}}^2}{2{\omega }^2}\left(1\pm \frac{{\omega }_B}{\omega }\right),$$(4)

where subscript L,R stands for left and right modes of circular polarisation. Throughout the rest of the paper, we will omit the subscript L,R. The approximation in Eq. (4) is valid under the condition ω_B≪ω. The trajectories of the rays can be determined by integrating Eq. (3). We considered a stationary, spherically symmetric plasma distribution with a Schwarzschild metric. Without loss of generality, we focus on the light ray confined to the plane of θ=π/2, the metric can be written as

$${\mathit{ds}}^2=-\left(1-\frac{2\mathit{GM}}{r}\right){\mathit{dt}}^2+{\left(1-\frac{2\mathit{GM}}{r}\right)}^{-1}{\mathit{dr}}^2+r^{2}d{\varphi }^2.$$(5)

Since g_αβ = 0 for α≠β, we can obtain the equation

$$\frac{{\mathit{dk}}_{\nu }}{d\tau }=\frac{1}{2}\left[g_{,\nu }^{\nu \nu }{k}_{\nu }^2+({\omega }_{\mathrm{e}}^2{)}_{,\nu }\pm ({\omega }_{\mathrm{e}}^2{\omega }_B/\omega {)}_{,\nu }\right].$$(6)

The last term in the equation represents the influence of the magnetic field, introducing anisotropy into the system. For wave propagating perpendicular to the magnetic field, i.e. B·k = 0, the magnetic field has no effect on the propagation of the wave. However, for waves aligned parallel or antiparallel to the magnetic field, the field will exert opposing effects on the propagation; this is a consequence of the distinct behaviours of the left and right hand mode polarisations. As light propagates, its direction changes due to the deflection, causing the ordinary and extraordinary waves to interchange along the path. This results in a varying contribution from B·k. In this study, we restricted our analysis to the small deflection angle and neglected this effect. We acknowledge that this approximation breaks down at a large deflection angle, such as those encountered near black holes or neutron stars. Thus, the scope of this work is limited to weak-field scenarios, and we assume that the magnetic field remains either parallel or antiparallel to the wave vector throughout the propagation. For high frequencies, two mode waves propagate approximately along the same rays, but with distinct speeds. This approximation is crucial for the phenomenon of Faraday rotation. In this work, we investigated the subtle differences in the ray paths of the two modes and the corresponding effects.

We used eⁱ and e_i for the unit vectors. The components of the vector k^ν, in a non-homogeneous medium, can be written as

$$\begin{array}{cc}\hfill k^{\nu }& =(\omega ,n\omega e^i),\hfill \\ \hfill k_{\nu }& =(-\omega ,n\omega e_i).\hfill \end{array}$$(7)

With the small angle approximation, we can select the z-axis for the trajectory of the light. From Eq. (6),

$$\frac{d{e}_{\nu }}{\mathit{dz}}=\frac{1}{2}\left(g_{33,\nu }+\frac{1}{n^2}{g}_{00,\nu }-\frac{({\omega }_{\mathrm{e}}^2{)}_{,\nu }\pm ({\omega }_{\mathrm{e}}^2{\omega }_B/\omega {)}_{,\nu }}{n^2{\omega }^2}\right),$$(8)

where ν = 1,2. We further assumed that the plasma distribution around the lens is spherically symmetric, i.e. n_e(r). This assumption can be relaxed to axial symmetry if necessary. By introducing the impact parameter b –defined as the separation between the lens and the photon trajectory– the position of the photon can be characterised by b and z along the line of sight, such that $r=\sqrt{b^2+z^2}$. The deflection angle is then derived by an integral along the direction of the ray e_α (Broderick & Blandford 2003; Bisnovatyi-Kogan & Tsupko 2010), and $\hat{\alpha }=e_{\alpha }(+\infty )-e_{\alpha }(-\infty )$ and

$$\hat{\alpha }=\frac{1}{2}{\displaystyle {{\displaystyle \int }}_{-\infty }^{\infty }}\mathit{dz}\frac{b}{r}\left[g_{33,r}+\frac{1}{n^2}{g}_{00,r}-\frac{K_{\mathrm{e}}{n}_{\mathrm{e},r}}{n^2{\omega }^2}\pm \frac{K_{\mathrm{e}}{K}_b(n_{\mathrm{e}}B{)}_{,r}}{n^2{\omega }^3}\right].$$(9)

Here, we used ω_B=eB(r)/(m_e)≡K_bB(r).

The first term represents the gravitational deflection. The second accounts for the deflection caused by a homogeneous plasma, which is typically small and will be neglected in this work. After reorganising the equation, we express the total deflection angle as a combination of three parts

$$\hat{\alpha }={\hat{\alpha }}_{\mathit{gl}}+{\hat{\alpha }}_{\mathit{pl}}\pm {\hat{\alpha }}_B,$$(10)

$$\approx -\frac{2R_s}{b}-\frac{K_{\mathrm{e}}}{2}{\displaystyle \int }\frac{n_{\mathrm{e},\alpha }}{{\omega }^2-{\omega }_{\mathrm{e}}^2}\mathit{dz}\pm \frac{K_{\mathrm{e}}{K}_b}{2}{\displaystyle \int }\frac{(n_{\mathrm{e}}B{)}_{,\alpha }}{{\omega }^3}\mathit{dz},$$(11)

where ${\hat{\alpha }}_{\mathit{gl}}$ and ${\hat{\alpha }}_{\mathit{pl}}$ represent the deflection angle caused by gravity and the density gradient of the plasma (i.e. the refraction), respectively. The third term, ${\hat{\alpha }}_B$, is a deflection generated by the gradient of the magnetic field. Usually, α_B<α_pl≪α_gl in lensing systems, and α_B is negligible. However, we retain this term in our analysis since it introduces a critical distinction between the two polarisation modes. Higher order contributions arise from Eq. (8), which, caused by the variations in the observational frequency along the line of sight as we discussed, are omitted for simplicity. Furthermore, we expand the deflection angle in the final term of Eq. (11) to facilitate our analysis:

$$\widehat{{\alpha }_B}=\frac{K_{\mathrm{e}}{K}_b}{2}{\displaystyle \int }\left(\frac{n_{\mathrm{e},\alpha }B+n_{\mathrm{e}}{B}_{,\alpha }}{{\omega }^3}\right)\mathit{dz}.$$(12)

The first contribution arises from variations in the electron density, while the second one from variations of the magnetic field. Thus, even if either the electron density or magnetic field remains constant – an unlikely scenario – there will still be a difference between the two polarisation modes.

2.2. Warm plasma

For the warm plasma, the dispersion relation will be changed by the temperature of the plasma (Broderick & Blandford 2003),

$$D=\left(1+\frac{1}{3}\frac{\mathit{kT}{\omega }_{\mathrm{e}}^2}{m_{\mathrm{e}}{\omega }^2}\right)k^{\mu }{k}_{\mu }+{\omega }_{\mathrm{e}}^2\pm {\omega }_{\mathrm{e}}^2\frac{{\omega }_B}{\omega },$$(13)

where T is the temperature of a thermal electron distribution. The alternation in the deflection properties is mainly governed by the temperature of the ionised gas. The high-temperature gas (i.e. ∼kev) can emit substantial high-energy radiation, which is directly observable. Thus, if we consider the case where the observational frequency exceeds the plasma frequency, the thermal modification for the cold or warm plasma is found to be less than 0.1%. Under such conditions, this effect can safely be neglected.

2.3. General cold plasma

For the general case, the dispersion relation becomes more complex and incorporates the covariant form of the magnetic field (Broderick & Blandford 2003). Here, we only discuss the dispersion relation in general, but we limit it to the approximations that we mentioned previously:

$$\begin{array}{cc}\hfill D& =k_{\mu }{k}^{\mu }-\delta {\omega }^2-\frac{\delta }{2(1+\delta )}\{\left[{\omega \prime }_B^2-(1+2\delta ){\omega }_B^2\right]\hfill \\ \hfill & \pm \sqrt{{\omega \prime }_B^4+2(2{\omega }^2-{\omega }_B^2-{\omega }_{\mathrm{e}}^2){\omega \prime }_B^2+{\omega }_B^4}\},\hfill \end{array}$$(14)

where

$$\delta =\frac{{\omega }_{\mathrm{e}}^2}{{\omega }_B^2-{\omega }^2},{\omega }_B^{\prime }=\frac{e{\mathcal{B}}^{\mu }{k}_{\mu }}{m_{\mathrm{e}}\omega }.$$(15)

Here, ${\omega }_B^{\prime }$ differs from ω_B, especially when light propagates in a strong gravitational field. We rewrite Eq. (14) in a compact form:

$$D=k_{\mu }{k}^{\mu }-\delta {\omega }^2-\frac{\delta }{2(1+\delta )}\left[A\pm \sqrt{C}\right],$$(16)

where

$$A={\omega \prime }_B^2-(1+2\delta ){\omega }_B^2,$$(17)

$$C=({\omega \prime }_B^2-{\omega }_B^2{)}^2+2(2{\omega }^2-{\omega }_{\mathrm{e}}^2){\omega \prime }_B^2.$$(18)

The parameter C characterises the splitting of the two polarisation modes. The prefactor $\frac{\delta }{2(1+\delta )}$ reveals an intriguing behaviour: when the magnetic field becomes sufficiently strong (${\omega }_B^2>{\omega }^2-{\omega }_{\mathrm{e}}^2$), the two modes flip. For instance, in the case of a double Einstein ring formed by gravitational lensing with a strong magnetic field, an increase in magnetic-field strength can cause the ordinary mode to transition from the inner ring to the outer ring, or vice versa. However, as we demonstrate later, the conditions required for such image splitting demand an extremely strong magnetic field, making this phenomenon unlikely to be observed in reality.

We further evaluated and simplified the parameters A and C within a Schwarzschild metric, starting from the frequency by

$${\omega }_B^{\prime }\approx \frac{\mathit{eB}}{m_{\mathrm{e}}(1-2M/r)},$$(19)

wherein only the radial component of the magnetic field is considered. In the event of a strong gravitational field – for example near a pulsar or a black hole, i.e. r∼10r_g – the discrepancy between ω_B and ${\omega }_B^{\prime }$ can reach ∼10%. In contrast, in typical gravitational lensing scenarios, where the deflection angle measures of the order of ∼1−10 arcsec, and r>10⁴r_g, the discrepancy becomes negligible. In this study, we focused on weak gravitational fields, which were thus approximated as ${\omega }_B^{\prime }$ by ω_B. The dispersion relation can then be written as

$$D=k_{\mu }{k}^{\mu }-\delta {\omega }^2+\frac{\delta }{2(1+\delta )}\left[2\delta {\omega }_B^2\mp \sqrt{2(2{\omega }^2-{\omega }_{\mathrm{e}}^2){\omega }_B^2}\right].$$(20)

This relation reduces to Eq. (2) in the limit of a weak magnetic field.

3. Toy models in weak magnetic field

To illustrate the effect of lensing on polarisation, we begin by examining a simple case where the wave vector is parallel to the magnetic field. To understand how lensing modifies polarisation, we construct the basic equation of lensing, incorporating the influence of the magnetic field. Our approach follows the general framework of lensing, which includes contributions from both gravitational and plasma effects (e.g. Schneider 2006; Tuntsov et al. 2016; Er & Mao 2022). Analogously to lensing in a vacuum, we construct a lens potential and calculate the time delay, which consists of both geometric and potential/dispersion contributions. The lens equation connects the angular positions between the image position θ and source position β by the reduced deflection angle α:

$$\beta =\theta -\alpha (\theta )=\theta -{\nabla }_{\theta }{\psi }_g(\theta )-{\nabla }_{\theta }{\psi }_{\mathit{pl}}(\theta )-{\nabla }_{\theta }{\psi }_B(\theta ),$$(21)

where ∇_θ is the gradient on the image plane, and ψ_g, ψ_pl is the effective lens potential for gravitational or plasma lensing, respectively. All the lensing distortions can be calculated from potential ψ. For example, the magnification produced by a lens is inversely proportional to the determinant of the Jacobian matrix, A, of the lens equation, μ⁻¹= det(A). The reduced deflection is given by $\alpha =\hat{\alpha }{D}_{\mathit{ds}}/D_s$, where we used the angular diameter distances between the lens and us, the source and us, and between the lens and the source as D_d,D_s,D_ds, respectively. We further define a time delay distance,

$$D_t=(1+z_d)\frac{D_d{D}_s}{D_{\mathit{ds}}},$$(22)

where z_d is the redshift of the lens. Then, the lens potential of the plasma and magnetic deflection can be written as

$${\psi }_{\mathit{pl}}+{\psi }_B=\frac{c}{D_t}\left(\frac{K_{\mathrm{e}}}{2{\omega }^2}\int n_{\mathrm{e}}\mathit{dz}\pm \frac{K_{\mathrm{e}}{K}_b}{2{\omega }^3}\int n_{\mathrm{e}}\mathit{Bdz}\right).$$(23)

In analogy to the Shapiro delay in the gravitational lensing, there is a delay of the signal with respect to that in the vacuum, which is dubbed ‘potential delay’ or ‘dispersion delay’:

$$t_{\mathrm{pot}}=\frac{D_t}{c}({\psi }_{\mathit{pl}}+{\psi }_B).$$(24)

The delay arises essentially from the change of propagation speed within the plasma medium. As this speed depends on frequency, the phenomenon is commonly described by the dispersion relation. The total time delay including the deflection (the geometric delay) is

$$t=\frac{D_t}{2c}{\alpha }^2+t_{\mathit{gl}}+t_{\mathit{pl}}\pm t_B.$$(25)

The geometric delay is usually negligible compared to the potential delay. However, in the case of strong lensing, the geometric delay leads to a significant contribution (Tsupko et al. 2020). Next, we calculated the delay difference between left and right modes of polarisation, as this is our primary interest:

$$\begin{array}{cc}\hfill \mathrm{\Delta }{t}_{\mathit{LR}}=& \frac{D_t}{2c}\left({\alpha }_L^2-{\alpha }_R^2\right)+2t_B\hfill \\ \hfill =& \frac{2D_t}{c}{\alpha }_{\mathit{gl}}{\alpha }_B+\frac{2D_t}{c}{\alpha }_{\mathit{pl}}{\alpha }_B+\frac{K_{\mathrm{e}}{K}_b}{{\omega }^3}\int n_{\mathrm{e}}\mathit{Bdz}.\hfill \end{array}$$(26)

The last term arises from the constant magnetic field and leads to Faraday rotation. We further assume that the left- and right-mode polarisations propagate along the same trajectory, implying that the two rays are coherent. Then, the phase difference can be converted to the rotation of the linear polarisation:

$$\begin{array}{cc}\hfill \mathrm{\Delta }\varphi & =2\pi \omega \mathrm{\Delta }{t}_{\mathit{LR}}\hfill \\ \hfill & =\frac{4\pi \omega D_t}{c}{\alpha }_{\mathit{gl}}{\alpha }_B+\frac{4\pi \omega D_t}{c}{\alpha }_{\mathit{pl}}{\alpha }_B+\frac{2\pi K_{\mathrm{e}}{K}_b}{{\omega }^2}\int n_{\mathrm{e}}\mathit{Bdz}.\hfill \end{array}$$(27)

Since α_gl is achromatic, the first term scales with λ², analogously to the wavelength dependence in Faraday rotation. The second term has a stronger dependence, scaling with λ⁴. While the deflection angles in both geometric terms are generally small, two geometric delays, which are proportional to the distance, D_t, become significant when the lens is located at cosmological distances. We compare the three terms in Table 1.

We adopted a singular isothermal sphere (SIS Binney & Tremaine 2008) for the mass model of the lens and demonstrate the magnetic effects in a galaxy-scale lens. A moderate redshift for the lens (z_d = 0.2) and a possible redshift for the background source (z_s = 0.5) is used, which is suggested by the peak redshift of the FRB (CHIME/FRB Collaboration 2021). We used a massive lens galaxy with σ_v = 300 km/s, which gives us an Einstein radius of θ_E∼1.1 arcsec. Similarly, for the density profile of the free electron we adopted a power law with a power index of h = 2 (e.g. Bisnovatyi-Kogan & Tsupko 2009; Er & Rogers 2019). For the magnetic field, we used the observational result from our Milky Way, which is about ∼10 μG on average (Beck & Wielebinski 2013; Han 2017). The Larmor frequency is on the level of ∼kHz with such a magnetic field. The profile of the magnetic field is a power law with a different index. We adopted two models in this section (h_b = 1,3). Such a choice was motivated by the observation of the central black hole in M87 (Kino et al. 2014; Event Horizon Telescope Collaboration 2021). We defined a scale radius of R₀ = 10 kpc; thus, we can write the profiles of electron density and magnetic field:

$$n_{\mathrm{e}}(r)=n_0(R_0/r{)}^2,B(r)=B_0(R_0/r{)}^{h_b},h_b=1,3,$$(28)

where we have n₀ = 0.01 cm⁻³ and B₀ = 10⁻⁵ Gauss. We followed the definition of the dispersion measure (DM) for the approximation of projected electron density,

$$\mathit{DM}(\theta )\approx N_{\mathrm{e}}(\theta )=\int n_{\mathrm{e}}(r)\mathit{dz},$$(29)

and rotation measure (RM) for the magnetic field,

$$\mathit{RM}=\frac{2r_{\mathrm{e}}^{2}c{ϵ}_0}{e}\int n_{\mathrm{e}}(r)B_{\parallel }d{z}_l.$$(30)

In our choice, we have DM = 100 pc cm⁻³ and RM = 810 rad m⁻² at R₀ = 10 kpc. The potential (dispersion) delay can be written by

$$t_{\mathit{pl}}=\frac{{\lambda }^2{r}_{\mathrm{e}}}{2\pi c}\mathit{DM}\pm \frac{{\lambda }^3}{2\pi c}\mathit{RM}.$$(31)

Then, the plasma lensing potential can be calculated from Eq. (24), as well as the deflection angle:

$${\alpha }_{\mathit{pl}}=-\frac{{\theta }_0^3}{{\theta }^2}\pm \frac{{\theta }_B^{h+1}}{{\theta }^h},$$(32)

where h = 2+h_b, the characteristic radius is (Bisnovatyi-Kogan & Tsupko 2010; Er & Rogers 2019)

$${\theta }_0^3=\frac{{\lambda }^2{r}_{\mathrm{e}}{n}_0}{\sqrt{\pi }}\frac{R_0^2}{D_t{D}_d}\frac{\mathrm{\Gamma }(3/2)}{\mathrm{\Gamma }(1)},$$(33)

and

$${\theta }_B^{h+1}=\frac{{\lambda }^3{r}_{\mathrm{e}}^{2}c{ϵ}_0{n}_0{B}_0}{\sqrt{\pi }e{D}_t}\frac{R_0^h}{D_d^{h-1}}\frac{\mathrm{\Gamma }(\frac{h+1}{2})}{\mathrm{\Gamma }(h/2)},$$(34)

where Γ(x) is the Gamma function.

We calculated the deflection angles caused by gravity α_gl, plasma α_pl, and magnetic field α_B separately. In a galaxy-scale strong lensing, α_gl is of the order of ∼1 arcsec. Given the electron density and magnetic field we adopted, α_pl is ∼10⁻⁴ arcsec, and α_B is ∼10⁻¹⁴ arcsec for the light at 1 GHz. Thus, the angular separation induced by the magnetic field is too small to be spatially resolved, rendering two images indistinguishable.

As mentioned above, the arrival time difference between the two modes is amplified by cosmic distance. In Fig. 1, we show Δt_LR, which is the time delay difference as a function of the lensed image position, θ. For both magnetic profiles, the delay difference caused by the gravitational deflection (red) is comparable to and greater than those induced by other effects. It exhibits a similar dependence on θ to the delay associated with Faraday rotation (blue). The delay caused by the plasma deflection, however, is relatively small, and decreases rapidly with increasing θ. In the second profile (h_b = 3), Δt_LR decreases with θ rapidly. While at small θ, the delay is large. Moreover, for the image formed at small θ, i.e. near the centre of the lens, the arrival-time difference is large enough compared to the width of the FRB (∼1 millisecond). It is possible to widen the shape of the pulse of the signal, or even split the pulse. In gravitational lensing, the images formed within the Einstein radius are usually demagnified. However, a high plasma density in the central region of the lens can mitigate the demagnification, potentially generating detectable images (Er & Mao 2022).

Fig. 1. Arrival-time difference between left- and right-mode polarisations. The shadow covers the frequency 0.5−5 GHz, and the blue shadow presents that due to different velocities of the two modes. The red (green) shadow presents that due to different paths of the propagation caused by gravitational deflection (plasma deflection). The top (bottom) panel is for the magnetic profile with an index of hb = 1 (hb = 3).

In Figs. 2 and 3, the rotations of the linear polarisation in a lensed image are shown. In the top panel, it is evident that the rotation is primarily driven by gravitational deflection. The bottom panel illustrates that the rotation due to gravitational deflection exhibits the same frequency dependence as Faraday rotation. Unlike the rotation induced by plasma deflection, which scales as ∝ν⁻⁴, the gravitational rotation cannot be distinguished from the Faraday rotation.

Fig. 2. Rotation of linear polarisation with image position (top, reference frequency ω = 0.5 GHz) and frequency (bottom, image position θ = 2 arcsec).

Fig. 3. Same as Fig. 2, but for hb = 3.

4. Strong magnetic field

In this section, we consider a case with a relatively strong magnetic field and apply the dispersion relation to Eq. (20). The Larmor frequency depends on the relative orientation between the light and the magnetic field, which influences the deflection of the light. However, we continue to adopt the thin-lens approximation for simplicity.

We took the point-mass model as our example (i.e. a black hole), since it has been suggested that in order to launch a jet, a strong magnetic field (∼10³⁻⁴ G) is required near the BH (Blandford & Znajek 1977; Baczko et al. 2016; Johnson et al. 2015). We used a nearby supermassive BH, M87, as our lens, of which the mass is 6×10⁹ M_⊙ (Gebhardt & Thomas 2009) and the distance is 16.7 Mpc (Jordán et al. 2005). The magnetic-field model was taken to be a radial power law (1/r) again (Petterson 1974). To better demonstrate the effect of the magnetic field, we took a high electron density, n_e = 100 cm⁻³, and a strong magnetic field, B = 10 G at R₀ = 100r_g, which agrees with the observation (Kino et al. 2014). The Larmor frequency approaches ∼0.1 GHz, or even higher in the centre of the lens.

The lensing properties of a point-mass model with plasma were studied in Tsupko & Bisnovatyi-Kogan (2020), which discusses the hill and hole structures of the magnification curve. Here, we compare the difference of the two modes of light. The deflection angle by gravity of a point lens is

$${\alpha }_{\mathit{gl}}=\frac{{\theta }_E^2}{\theta },\mathrm{with}{\theta }_E^2=\frac{4\mathit{GM}}{c^2}\frac{D_{\mathit{ds}}}{D_d{D}_s},$$(35)

and M is the total mass of the lens. Here, the mass of the free electron is not included, as they will not cause a significant effect. To further simplify the mathematics, we used the source distance with D_s = 2D_d. The Einstein radius of the point gravitational lens is θ_E≈1.2 milliarcsec.

We calculated the total deflection angles for both polarisation modes of light (Fig. 5). Both deflection angles are smaller than those in a vacuum. The difference between the two modes is small, but it becomes slightly larger at small θ. While the gravitational deflection in vacuum diverges at the centre of the lens, the deflection angles in the presence of plasma remain finite. In the bottom panel, β_gl increases monotonically with θ. The combined effects of plasma magnetic deflections alter the behaviours of lensed images near the centre, resulting a pronounced turnover in β_±. This turnover is a direct consequence of the deflection induced by the magnetised plasma. We compute the image splitting between the two polarisation modes, i.e. the difference in the deflection angle, Δα, at θ = 1 milliarcsec. In Fig. 4, the grey shading represents Δα on a logarithmic scale. The red line indicates the condition where the images will flip between the two polarisation modes. Using 0.1 milliarcsec as the limit for spatial resolution (purple curve), it is clear that the possibility of detecting two images with different polarisations exists only near the centre of the BH. For this work, we made calculations at an observational frequency of 1 GHz and exclude image splitting unless otherwise specified.

Fig. 4. Maximum difference of deflection angle between two polarisation modes. The point-mass model in Sect. 4 is adopted. The red line shows the condition of image flip. The yellow, green, and purple curves mark Δα = 10−2,10−3, and 10−4 arcsec, respectively.

Fig. 5. Top: Deflection angle of point lens with strong magnetic plasma. The red (blue) curve presents the ordinary (extraordinary) mode of the wave. Bottom: Relation between source position, β, and image position, θ.

In Fig. 6, we compare the magnification curves for a lens in a magnetic plasma with those in a vacuum, with a focus on the central region of the lens. The critical radius (where μ→∞) is reduced for both polarisation modes. Moreover, both modes exhibit extra hill and hole structure in the inner region, resembling the result in plasma lensing (e.g. Tsupko & Bisnovatyi-Kogan 2020). These structures imply the potential formation of multiple ‘Einstein rings’, although detecting them would require exceptionally high resolution. We show the arrival time difference between the two polarisation modes in Fig. 7. The blue shading represents the delay caused by the differing velocities of the two modes, while the red shading indicates the delay due to the difference in their trajectories. An interesting point is that for images formed close to the centre of the lens, the time difference reverses. This reversal can be interpreted as an inversion in dispersion relation, as we discussed in Eq. (16). In this scenario, the trajectory difference becomes the dominant factor influencing the delay time. Therefore, the polarisation information is determined by both the deflection of the signal and the magnetic field. However, the time delays presented here are calculated based on the positions of lensed images. In reality, the lensed images corresponding to the two modes originate from slightly different source positions. This discrepancy becomes significant in the presence of a strong magnetic field. Although we did not consider the image splitting in this work, we provide an example of lensed images for two polarisation modes in Fig. 8. For the ordinary mode (+mode), a ‘double Einstein ring’ can form. In the top middle panel, we can also see that an extra faint image can form on the opposite side of the lens.

Fig. 6. Magnification curves of point lens with magnetic plasma. The dashed green curve presents the magnification in vacuum. The red (blue) one shows that of the ordinary (extraordinary) mode. The vertical cyan line marks the position of the Einstein radius in vacuum.

Fig. 7. Similar to Fig. 1; red represents arrival-time difference between the two modes of the wave caused by the different paths of the light propagation (textraordinary−tordinary). The shadows cover a frequency range from 1 to 5 GHz.

Fig. 8. Demonstration of simulated lensed images. The red plus in the centre marks the position of the lens. Top: Only gravitational deflection. Middle: Deflection with magnetic plasma for extraordinary mode. Bottom: Deflection with magnetic plasma for ordinary mode. The Einstein radius of the point lens is ∼1.2 milliarcsec. The relative position of the background source with respect to the lens is 0,0.5,1,1.5,2 milliarcsec from left to right, respectively.

For the magnetic profile with power index h_b = 3, similar lensing behaviour can be found, but it rapidly decreases with θ. Complex lensing effects will appear near the centre of the lens. Such a strong magnetic field can only usually be found near a neutron star or black hole, as shown by the Event Horizon Telescope near M87 (Event Horizon Telescope Collaboration 2021). The observation of M87 may not be dramatically affected by lensing effects since the lensing distance D_t is small. The magnetic field and its gradient are strong and may affect the Faraday rotation relation.

5. Summary and discussions

In this work, we studied the gravitational lensing embedded in a magnetised plasma. The magnetised medium introduces additional deflections and exhibits birefringence. In a uniform magnetic field, the two polarisation modes propagate along an identical path, but with different arrival times or phases, which causes the Faraday rotation of the linear polarisation. However, in the presence of magnetic-field gradients, the trajectories of the two polarisation modes diverge. The distinction of the light path is further enhanced by the gravitational field, resulting in significant time delays (which we called geometric delay) and additional effects, such as rotation of the linear polarisation. First, we compared the two modes in a galaxy-scale lens with a weak magnetic field. Geometric delay and geometric rotation show the same dependence on observational frequency as Faraday rotation, and their magnitudes are greater. This lensing-induced geometric rotation occurs generically in the presence of magnetic-field gradient and plasma density gradient. For an order of magnitude estimate, only for strongly lensed sources – i.e. deflection angle α∼1 arcsec, at cosmic distance, D_t∼100 Mpc – is the geometric rotation comparable with or even greater than the Faraday rotation. In the second example, we considered a point lens with a strong magnetic field. The lensed images in plasma differ from those in vacuums in several ways. The deflection angle is smaller in a plasma medium than that in vacuum. The hill and hole structures can appear in the magnification curve, varying between the two polarisation modes. The lensed images can split for the two modes – displaying complex behaviours, especially near the centre of the lens – although extremely high resolution is required to resolve this. For short-period radio sources, such as FRBs, the distinct time delays between the two modes may alter the pulse structure, potentially leading to multiple pulses with different polarisation modes.

The study presented here is based on several approximations in the calculations, and requires further investigations. First of all, the refraction index employed is a scaler and thus insufficient to describe all propagation modes of the light, especially under the conditions involving large deflection angles. In the case of a strong gravitational field, for example near a black hole, the small angle approximation breaks down. The lens equation describing the deflection in the strong field incorporating a tensor-mode dispersion relation is necessary to model these effects. The photon frequency will be changed by the strong gravitational field, and it also affects the results in this work. Additionally, while the thin-lens approximation adopted in this study is adequate for our example of a galaxy lens, it requires modification when applied to more complex scenarios.

The temperature of the plasma in the interstellar medium or intergalactic medium may not be a critical factor affecting the results in this study. The situation will not be the same when we look at the vicinity of a BH. The deflection can be complicated because of strong gravity and the high-temperature gas. Moreover, since the difference between the two modes is small, other effects could influence the result, such as the motion of the lens and source, the structures along the line of sight, the variations and small structures in plasma density, and so on. Nevertheless, polarisation measurements offer valuable additional constraints on both the lens system and the properties of the background source itself. The propagation effects, particularly in regions influenced by strong gravitational field, are non-negligible and must be explicitly accounted for.

Acknowledgments

I thank the referee for constructive and valuable comments. I also like to thank Oleg Yu Tsupko for valuable suggestions and comments on the work.

References

All Tables

All Figures

Fig. 1. Arrival-time difference between left- and right-mode polarisations. The shadow covers the frequency 0.5−5 GHz, and the blue shadow presents that due to different velocities of the two modes. The red (green) shadow presents that due to different paths of the propagation caused by gravitational deflection (plasma deflection). The top (bottom) panel is for the magnetic profile with an index of hb = 1 (hb = 3).

In the text

	Fig. 2. Rotation of linear polarisation with image position (top, reference frequency ω = 0.5 GHz) and frequency (bottom, image position θ = 2 arcsec).
In the text

	Fig. 3. Same as Fig. 2, but for hb = 3.
In the text

	Fig. 4. Maximum difference of deflection angle between two polarisation modes. The point-mass model in Sect. 4 is adopted. The red line shows the condition of image flip. The yellow, green, and purple curves mark Δα = 10−2,10−3, and 10−4 arcsec, respectively.
In the text

	Fig. 5. Top: Deflection angle of point lens with strong magnetic plasma. The red (blue) curve presents the ordinary (extraordinary) mode of the wave. Bottom: Relation between source position, β, and image position, θ.
In the text

	Fig. 6. Magnification curves of point lens with magnetic plasma. The dashed green curve presents the magnification in vacuum. The red (blue) one shows that of the ordinary (extraordinary) mode. The vertical cyan line marks the position of the Einstein radius in vacuum.
In the text

	Fig. 7. Similar to Fig. 1; red represents arrival-time difference between the two modes of the wave caused by the different paths of the light propagation (textraordinary−tordinary). The shadows cover a frequency range from 1 to 5 GHz.
In the text

Fig. 8. Demonstration of simulated lensed images. The red plus in the centre marks the position of the lens. Top: Only gravitational deflection. Middle: Deflection with magnetic plasma for extraordinary mode. Bottom: Deflection with magnetic plasma for ordinary mode. The Einstein radius of the point lens is ∼1.2 milliarcsec. The relative position of the background source with respect to the lens is 0,0.5,1,1.5,2 milliarcsec from left to right, respectively.

In the text